as far as I know there are two common writings of Bassai/Passai in kanji. It would be better to write it in Katakana because the real meaning of Bassai/Passai is not known. The Kanji used were just taken because the pronounciation sounds like Bassai/Passai.

However when you look at the kanji that were used to write that than they mean the following.

Kanji one differs. Both mean to break, to remove, to crush, to open up.
Kanji two means obstruction, barrier or fortress

So the translation I prefer in order with the kanji writing is "to remove an obstruction".

In Shotkan's Secret Bruce Clayton makes the observation that the direct translation of the kanji is "To break into the fortress". He goes on to explain that based on the movements, the classical discussion, and the reasonable interpretation of the bunkai that it is more likely that they intended it to mean "to break out of the fortress". Now I'm paraphrasing here obviously, but Clayton makes his case by pointing out that the bunkai within bassai(dai) tends to be in close quarters, and that there are not really any techniques represented that show any offensive movements. This leads him to the conclusion that the kata represents fighting your way out of a situation, lending itself to the name of "breaking out" instead of in.

I really like the "remove obstacle" translation. This is a combative theme worthy of a Kata in my eyes. I also think that the movements (often intrepreted as blocks) lend themselves fine toward this way of translating the name.

It would be better to write it in Katakana because the real meaning of Bassai/Passai is not known. The Kanji used were just taken because the pronounciation sounds like Bassai/Passai.

This is a really important point. People should keep in mind that the kanji we use for many kata names today are relatively recent innovations that will have no connection to the original meaning of the name.

Funakoshi wrote (Karate-Do: My way of life), “… almost all the kata names I described in my book [Ryukyu Kenpo: Karate – 1922] were of Okinawan origin Pinan, Naihanchi, Chinto, Bassai, Seishan, JItte, Jion, Sanchin and the like. There were, in fact, the names I learnt long ago from my own teachers. No one, by now, had any idea how they came into being, and people found them difficult to learn … I began to give the kata names that were easier for Japanese people to use and that have now became familiar all over the world … I have no doubt that as times change, again and then again, the kata will be given new names. And that, indeed, is how it should be.”

Because no one had any idea how many of the kata names came into being, and because that are not Japanese in origin, it is unsurprising that the written sources use katakana (symbols that give sound but no meaning and that are used for foreign words) and opposed to kanji (characters that have meaning). It was later on that the kata had kanji attached to them. Kanji were picked that would give roughly the right sound, but any meaning attached to those characters has not been attached to the kata since its creation. Basically, history bequeathed us with the sound “Bassai / Passai” (originating from non-Japanese languages i.e. perhaps one of the Okinawan dialects) and people went “and that sounds a little like the sound these characters would make in Japanese”. No direct link though.

Some kata we do have the kanji for, but Passai / Bassai is not one of them. So as Holger said above, we really have no idea what “Bassai” means.

I forwarded this conversation onto my friend Marc Miyake, who happens to be a linguist specializing in Eastern Asian languages. Here's what he had to say:

"All this time I thought bassai was ?? (pull out + crush). I didn't know about the other spelling ?? (pull out + fortress).
I agree with Abernethy: one should not confuse *spellings* with *etymologies*.
Here's an English parallel: the word bridegroom has nothing to do with grooms of horses - the word guma 'man' was replaced by the unrelated word groom. Similarly, Ryukyuan roots could have been replaced by similar-sounding, unrelated Japanese roots.
I vaguely recall seeing some Chinese martial arts terms of obscure origin. They could be similar to bassai: i.e., they might be attempts to phonetically approximate 'characterless' words. Assigning characters to words makes them seem 'better' - hence the tendency to write kanji instead of 'mere' kana for kata names.
One can often tell that a Chinese character name is a phonetic approximation by looking at the meanings and seeing that they have nothing to do with the referent: e.g., Mandarin ?? Luoma 'net horse' for Rome. "Net horse' makes no sense so it must be a name.
But the trouble with ?? (pull out + crush) and ?? (pull out + fortress) for bassai is that they sound like they might be meaningful, though I'll take Abernethy's word that they're not. (The existence of variants could be a warning sign - if bassai really meant one or the other, the other spelling might not exist.)"